Agreed. IMO the biggest problem with online communication is that people don't realize how much ambiguity there is in language, or how easy it is to misread what people type into little boxes on web pages.
Every tweet is like an underdetermined system of equations with infinitely many solutions (solution == what they really meant when they typed up the tweet, and everything else that readers project onto posters). Everyone has to choose how to interpret ambiguous statements all the time, and by nature our brains are more likely to choose interpretations that conform to our own expectations.
We can create our own private echo chambers if we aren't careful, where we always interpret what people who are on "our side" say in the most charitable way possible, and we always interpret statements from opponents in terms of the biggest strawmen our brains can come up with. This is a pitfall of any sort of communication, but it seems to be a vastly greater danger with online communication.
People tend to blame the lousiness of online communication on other people, and there's some truth to that, but I think it's just inherent in both the medium and human nature. In the long term I hope that digital mediums can evolve to support better communication and provide more fertile ground for more constructive discussions.